A274338 -id:A274338 - cp's OEIS frontend

A187793 Sum of the deficient divisors of n.

1, 3, 4, 7, 6, 6, 8, 15, 13, 18, 12, 10, 14, 24, 24, 31, 18, 15, 20, 22, 32, 36, 24, 18, 31, 42, 40, 28, 30, 36, 32, 63, 48, 54, 48, 19, 38, 60, 56, 30, 42, 48, 44, 84, 78, 72, 48, 34, 57, 93, 72, 98, 54, 42, 72, 36, 80, 90, 60, 40, 62, 96, 104, 127, 84, 72, 68, 126, 96, 74, 72, 27

Offset: 1

Timothy L. Tiffin, Jan 06 2013

Sum of divisors d of n with sigma(d) < 2*d.
a(n) = sigma(n) when n is itself also deficient.
Also, a(n) agrees with the terms in A117553 except when n is a multiple (k > 1) of either a perfect number or a primitive abundant number.
Notice that a(1) = 1. The remaining fixed points are given by A125310. - Timothy L. Tiffin, Jun 23 2016
a(A028982(n)) is an odd integer.  Also, if n is an odd abundant number that is not a perfect square and n has an odd number of abundant divisors (e.g., 945 has one abundant divisor and 4725 has three abundant divisors), then a(n) will also be odd: a(945) = 975 and a(4725) = 2675. - Timothy L. Tiffin, Jul 18 2016

			a(12) = 10 because the divisors of 12 are 1, 2, 3, 4, 6, 12; of these, 1, 2, 3, 4 are deficient, and they add up to 10.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of divisors

Cf. A000203, A005100, A028982, A080226, A117553, A125310, A125499, A187794, A187795, A247328, A274338, A274339, A274340, A274380, A274549, A274829, A294886, A294934.

A187793 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if numtheory[sigma](d) < 2*d then
            a := a+d ;
        end if ;
    end do:
    a ;
end proc:# R. J. Mathar, May 08 2019

Table[Total@ Select[Divisors@ n, DivisorSigma[1, #] < 2 # &], {n, 72}] (* Michael De Vlieger, Jul 18 2016 *)

a(n)=sumdiv(n,d,if(sigma(d,-1)<2,d,0)) \\ Charles R Greathouse IV, Jan 07 2013

From Antti Karttunen, Nov 14 2017: (Start)
a(n) = Sum_{d|n} A294934(d)*d.
a(n) = A294886(n) + (A294934(n)*n).
a(n) + A187794(n) + A187795(n) = A000203(n).
(End)

a(54) corrected by Charles R Greathouse IV, Jan 07 2013

A274339 The period 3 sequence of the iterated sum of deficient divisors function (A187793) starting at 15.

Original entry on oeis.org

15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18

Offset: 1

Timothy L. Tiffin, Jun 22 2016

This sequence is generated in a similar way to aliquot sequences or sociable chains, which are generated by iterating the sum of proper divisors function (A001065). It appears to be the only one of period (order, length) 3 that A187793 generates under iteration.
If sigma(N) is the sum of positive divisors of N, then:
  a(n+1) = sigma(a(n)) if a(n) is a deficient number (A005100),
  a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number (A071395),
  a(n+1) = sigma(a(n))-a(n)-m if a(n) is an abundant number with one proper divisor m that is either perfect (A275082) or abundant, and so forth.
This is used in the example below.
A284326 also generates this sequence under iteration. - Timothy L. Tiffin, Feb 22 2022

			a(1) = 15;
a(2) = sigma(15) = 24;
a(3) = sigma(24) - 24 - 12 - 6 = 18;
a(4) = sigma(18) - 18 - 6 = 15 = a(1).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A001065, A005100, A005101, A071395, A125310, A187793, A274338, A274340, A274380, A274549, A275082, A284326.

LinearRecurrence[{0,0,1},{15,24,18},90] (* or *) PadRight[{},90,{15,24,18}] (* Harvey P. Dale, Aug 06 2023 *)

Vec(3*x*(5 + 8*x + 6*x^2) / ((1 - x)*(1 + x + x^2)) + O(x^40)) \\ Colin Barker, Jan 30 2020

a(n+3) = a(n).

G.f.: 3*x*(5 + 8*x + 6*x^2) / ((1 - x)*(1 + x + x^2)). - Colin Barker, Jan 30 2020

A274340 The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 19.

Original entry on oeis.org

19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36

Offset: 1

Timothy L. Tiffin, Jun 22 2016

This sequence is generated in a similar way to aliquot sequences or sociable chains, which are generated by iterating the sum of proper divisors function (A001065). It appears to be one of two sequences of period (order, length) 4 that A187793 generates under iteration. The other one is A274380.
If sigma(N) is the sum of positive divisors of N, then:
  a(n+1) = sigma(a(n)) if a(n) is a deficient number (A005100),
  a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number (A071395),
  a(n+1) = sigma(a(n))-a(n)-m if a(n) is an abundant number with one proper divisor m that is either perfect (A275082) or abundant, and so forth.
This is used in the example below.

			a(1) = 19;
a(2) = sigma(19) = 20;
a(3) = sigma(20) - 20 = 22;
a(4) = sigma(22) = 36;
a(5) = sigma(36) - 36 - 18 - 12 - 6 = 19 = a(1).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A001065, A005100, A005101, A071395, A125310, A187793, A274338, A274339, A274380, A274549, A275082.

PadRight[{},100,{19,20,22,36}] (* Paolo Xausa, Oct 16 2023 *)

Vec(x*(19 + 20*x + 22*x^2 + 36*x^3) / (1 - x^4) + O(x^80)) \\ Colin Barker, Jan 30 2020

a(n+4) = a(n).
a(n) = A187793(a(n-1)).
G.f.: x*(19 + 20*x + 22*x^2 + 36*x^3) / (1 - x^4). - Colin Barker, Jan 30 2020

A274380 The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 34.

Original entry on oeis.org

34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48

Offset: 1

Timothy L. Tiffin, Jun 22 2016

This sequence is generated in a similar way to aliquot sequences or sociable chains, which are generated by iterating the sum of proper divisors function (A001065). It appears to be one of two sequences of period (order, length) 4 that A187793 generates under iteration. The other one is A274340.
If sigma(N) is the sum of positive divisors of N, then:
  a(n+1) = sigma(a(n)) if a(n) is a deficient number (A005100),
  a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number (A071395),
  a(n+1) = sigma(a(n))-a(n)-m if a(n) is an abundant number with one proper divisor m that is either perfect (A275082) or abundant, and so forth.
This is used in the example below.
A284326 also generates this sequence under iteration. - Timothy L. Tiffin, Feb 22 2022

			a(1) = 34;
a(2) = sigma(34) = 54;
a(3) = sigma(54) - 18 - 6 = 42;
a(4) = sigma(42) - 42 - 6 = 48;
a(5) = sigma(48) - 48 - 24 - 12 - 6 = 34 = a(1);
  :
  :

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A001065, A005100, A005101, A071395, A125310, A187793, A274338, A274339, A274340, A274549, A275082, A284326.

Vec(2*x*(17 + 27*x + 21*x^2 + 24*x^3) / ((1 - x)*(1 + x)*(1 + x^2)) + O(x^80)) \\ Colin Barker, Jan 30 2020

a(n+4) = a(n).

G.f.: 2*x*(17 + 27*x + 21*x^2 + 24*x^3) / ((1 - x)*(1 + x)*(1 + x^2)). - Colin Barker, Jan 30 2020

A274549 Numbers found in the cycles of the iterated sum of deficient divisors function.

Original entry on oeis.org

1, 6, 15, 18, 19, 20, 22, 24, 28, 34, 36, 42, 48, 52, 54, 76, 78, 84, 90, 98, 140, 171, 260, 308, 336, 496, 8128, 33550336, 8589869056

Offset: 1

Timothy L. Tiffin, Jun 27 2016

The 1-cycles (or fixed points) greater than 1 are given in A125310, the 3-cycle terms are given in A274339, the 4-cycle terms are given in A274340 and A274380, and the 10-cycle terms are given in A274338. This sequence was suggested to me by Michel Marcus when I was submitting the 3-cycle, 4-cycle, and 10-cycle sequences.

Cf. A125310, A187793, A274338, A274339, A274340, A274380.

cp's OEIS Frontend

A187793 Sum of the deficient divisors of n.

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A274339 The period 3 sequence of the iterated sum of deficient divisors function (A187793) starting at 15.

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A274340 The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 19.

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A274380 The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 34.

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A274549 Numbers found in the cycles of the iterated sum of deficient divisors function.

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